Sequential Fixed-Width Confidence Interval for the Product of Two Means

For the product of two population means, the problem of constructing a fixed-width confidence interval with preassigned coverage probability is considered. It is shown that the optimal sample sizes which minimize the total sample size and at the same time guarantee a fixed-width confidence interval of desired coverage depend on the unknown parameters. In order to overcome this, a fully sequential procedure consisting of a sampling scheme and a stopping rule are proposed. It is then shown that the sequential confidence interval is asymptotically consistent and the stopping rule is asymptotically efficient, as the width goes to zero. Furthermore, a second order result for the difference between the expected stopping time and the (total) optimal fixed sample size is established. The theoretical results are supported by appropriate simulations.     

Sequential Estimation of the Maximum in a Model for Corrosion Data

One method of monitoring corrosion in an underground storage tank involves placing a sensor in the tank and running it around the tank's interior. As it runs, the sensor records the local thickness of the tank. In this paper we consider the problem of estimating the maximum pit depth by providing a confidence interval that achieves both a specified confidence level and a specified degree of precision. A particular model, the three-parameter beta, is considered, and a stopping rule for determining the sample size is proposed. It is shown that the stopping rule achieves the desired confidence level and precision, asymptotically as the precision requirement becomes increasingly stringent. Moreover, the stopping rule is asymptotically efficient in terms of sample size. The limiting distribution of the stopping rule is derived, and simulation results are presented to supplement the asymptotics with finite sample size behavior.     

渐近最优的经验贝叶斯决策

本文获得了刻度指数族变量带误差情形下的贝叶斯决策,且利用解卷积的核方法构造出了经验贝叶斯决策.在适当的条件下,证明了经验贝叶斯决策的渐近最优性.     

Testing the Gumbel Hypothesis Via the Pot-Method

We consider an i.i.d. sample, generated by some distribution function, which belongs to the domain of attraction of an extreme value distribution with unknown shape and scale parameters. We treat the scale parameter as a nuisance parameter and establish for the hypothesis of Gumbel domain of attraction an asymptotically optimal test based on those observations among the sample, which exceed a given threshold sequence. Asymptotic optimality is achieved along certain contiguous extreme value alternatives within the concept of local asymptotic normality (LAN). Adaptive test procedures exist under restrictive assumptions. The finite sample size behavior of the proposed test is studied by simulations and it is compared to that of a test based on the sample coefficient of variation.     

Asymptotic vulnerability of positional voting rules to coalitional manipulation

Voting rule performances are sometimes evaluated according to their respective resistances to allow profitable misrepresentation of individual preferences. This seems to be a hard task when scoring systems with possibly non integer weights are involved. In this paper, it is shown how one can still obtain asymptotic results in these settings. Our analysis for three-candidate elections provides a characterization of unstable voting situations at which a positional voting rule is manipulable by some coalition not larger than an arbitrary proportion of the electorate. This allows us to address a conjecture by Pritchard and Wilson (2007). That is, under the Impartial Anonymous Culture (IAC), the plurality rule asymptotically minimizes the vulnerability to coalitional manipulation when the size of the manipulating coalition is unrestricted. This later result is no longer valid when only manipulation by small coalitions is considered: now, the Borda rule tends to outperform other rules. Furthermore, the vulnerability of a positional voting rule to coalitional manipulation is not affected by increasing the size of the manipulating coalition from 0.5 to 1.     

舍入数据下Lomax分布形状参数的经验Bayes检验问题

本文研究了舍入数据下Lomax分布形状参数的经验Bayes (EB)单侧检验问题.利用密度函数的递归核估计构造了参数的EB检验函数,并在适当的条件下证明了所提出的EB检验函数的渐近最优性,获得了它的收敛速度.最后,给出一个有关本文主要结果的例子.     

Safe scheduling: Setting due dates in single-machine problems

We consider single-machine stochastic scheduling models with due dates as decisions. In addition to showing how to satisfy given service-level requirements, we examine variations of a model in which the tightness of due-dates conflicts with the desire to minimize tardiness. We show that a general form of the trade-off includes the stochastic E/T model and gives rise to a challenging scheduling problem. We present heuristic solution methods based on static and dynamic sorting procedures. Our computational evidence identifies a static heuristic that routinely produces good solutions and a dynamic rule that is nearly always optimal. The dynamic sorting procedure is also asymptotically optimal, meaning that it can be recommended for problems of any size.     

Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses

The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.     

Asymptotically periodic solution and optimal harvesting policy for Gompertz system

In this paper, the single species modelled by (asymptotically) periodic Gompertz equation is investigated. It is shown that the (asymptotically) periodic system has a unique (asymptotically) periodic solution which is globally asymptotically stable for the positive solution. When the nonautonomous Gompertz equation is subject to harvesting, we study the optimal harvesting policy for the periodic system and obtain the corresponding optimal population level and the maximum sustainable yield. Further, when the functions in the exploited Gompertz system are stably bounded functions, we study the ultimately optimal harvesting policy. By choosing the average limiting maximum sustainable yield as management objective, the corresponding optimal population level is determined.     

A third order optimum property of the ML estimator in a linear functional relationship model and simultaneous equation system in econometrics

Summary The maximum likelihood (ML) estimator and its modification in the linear functional relationship model with incidental parameters are shown to be third-order asymptotically efficient among a class of almost median-unbiased and almost mean-unbiased estimators, respectively, in the large sample sense. This means that the limited information maximum likelihood (LIML) estimator in the simultaneous equation system is third-order asymptotically efficient when the number of excluded exogenous variables in a particular structural equation is growing along with the sample size. It implies that the LIML estimator has an optimum property when the system of structural equations is large. The research was partly supported by National Science Foundation Grant SES 79-13976 at the Institute for Mathematical Studies in the Social Sciences, Stanford University and Grant-in-Aid 60301081 of the Ministry of Education, Science and Culture at the Faculty of Economics, University of Tokyo. This paper was originally written as a part of the author's Ph.D. dissertation submitted to Stanford University in August, 1981. Some details of the paper were deleted at the suggestion of the associate editor of this journal.     

Sequential procedures for a class of distributions related to the uniform

Summary The main purpose of this paper is to examine some sequential inference procedures, mainly in the one-sided hypothesis testing situation, for the parameter of a class of distributions related to the Uniform distribution on (0, θ). The procedures are all based on the sequence of maxima of independent and identically distributed random variables and, for the case of a single (upper) boundary, procedures which are optimal in the sense of minimizing the average sample size are discussed. The impossibility of using two boundaries is demonstrated, thus leaving the best procedure with one boundary as the optimal procedure. The novelty of this procedure is that it uniformly minimizes the average sample size.     

随机效应模型中方差分量的经验Bayes检验问题

给出了双向分类随机效应模型中方差分量的Bayes检验的判决函数,利用核估计的方法,构造了相应的经验Bayes(EB)检验的判决函数．在适当的条件下证明了EB判决函数是渐近最优的且有收敛速度．给出了模型的特例和推广．最后,举出一个满足定理条件的例子．     

Asymptotic Optimality of Certain Multihypothesis Sequential Tests: Non‐i.i.d. Case

It is known that certain combinations of one‐sided sequential probability ratio tests are asymptotically optimal (relative to the expected sample size) for problems involving a finite number of possible distributions when probabilities of errors tend to zero and observations are independent and identically distributed according to one of the underlying distributions. The objective of this paper is to show that two specific constructions of sequential tests asymptotically minimize not only the expected time of observation but also any positive moment of the stopping time distribution under fairly general conditions for a finite number of simple hypotheses. This result appears to be true for general statistical models which include correlated and non‐homogeneous processes observed either in discrete or continuous time. For statistical problems with nuisance parameters, we consider invariant sequential tests and show that the same result is valid for this case. Finally, we apply general results to the solution of several particular problems such as a multi‐sample slippage problem for correlated Gaussian processes and for statistical models with nuisance parameters. This revised version was published online in June 2006 with corrections to the Cover Date.     

φ混合样本下帕累托分布参数的经验贝叶斯分析

在弱平稳φ混合样本下,讨论了Pareto分布参数的经验Bayes（EB）单边检验问题,构造了参数的经验Bayes检验函数,证明了它具有渐近最优（a.o.）性,并且获得了收敛速度。     

Bounding the L1 Distance in Nonparametric Density Estimation

Let X1, X2, ..., Xn be i.i.d. random variables with common unknown density function f. We are interested in estimating the unknown density f with bounded Mean Integrated Absolute Error (MIAE). Devroye and Gyrfi (1985, Nonparametric Density Estimation: The L1 View, Wiley, New York) obtained asymptotic bounds for the MIAE in estimating f by a kernel estimate fn. Using these bounds one can identify an appropriate sample size such that an asymptotic upper bound for the MIAE is smaller than some pre-assigned quantity w > 0. But this sample size depends on the unknown density f. Hence there is no fixed sample size that can be used to solve the problem of bounding the MIAE. In this work we propose stopping rules and two-stage procedures for bounding the L1 distance. We show that these procedures are asymptotically optimal in a certain sense as w 0, i.e., as one requires increasingly better fit.     

Empirical Bayes Test for the Parameter of Rayleigh Distribution with Error of Measurement

For the data with error of measurement in historical samples, the empirical Bayes test rule for the parameter of Rayleigh distribution is constructed, and the asymptotically optimal property is obtained. It is shown that the convergence rate of the proposed EB test rule can be arbitrarily close to O（n-1/2） under suitable conditions.     

基于纵向数据下连续型单参数指数族参数的经验贝叶斯检验(英文)

本文讨论了在纵向数据下,运用非参数估计方法构造了连续型单参数指数族参数的经验贝叶斯检验函数,证明了所提出的经验贝叶斯检验函数的渐近最优性,并获得了它的收敛速度.     

Limit Theorems for Risk Estimate in Models with Non-Gaussian Noise

The problem of constructing an estimate of a signal function from noisy observations, assuming that this function is uniformly Lipschitz regular, is considered. The thresholding of empirical wavelet coefficients is used to reduce the noise. As a rule, it is assumed that the noise distribution is Gaussian and the optimal parameters of thresholding are known for various classes of signal functions. In this paper a model of additive noise whose distribution belongs to a fairly wide class, is considered. The mean-square risk estimate of thresholding is analyzed. It is shown that under certain conditions, this estimate is strongly consistent and asymptotically normal.     

A Posteriori Error Estimators of Gradient Recovery Type for Fem of a Model Optimal Control Problem

In this paper, we derive an a posteriori error estimator of gradient recovery type for a model optimal control problem. We show that the a posteriori error estimator is equivalent to the discretization error in a norm of energy type on general meshes. Furthermore, when the solution of the control problem is smooth and the meshes are uniform, it is shown to be asymptotically exact.     

OPTIMAL HARVESTING TIME OF FARMED AQUATIC POPULATIONS WITH NONLINEAR SIZE‐HETEROGENEOUS GROWTH

Abstract This paper examines the question of optimal harvesting time in a size‐heterogeneous farmed aquatic population, using a model reflecting the effect of population density on both overall mortality rate and individual growth. This analysis enables an optimal harvesting rule to be deduced. The results obtained are applied to shrimp culture in recirculation systems in Mexico. Numerical solutions are derived for different production scenarios. Assuming identical culture conditions, results are also obtained under the hypothesis of homogeneous population growth, the view traditionally taken in the relevant economic literature. The optimal harvesting times calculated tend to decrease with higher densities, although this rule fails under the size‐heterogeneous population model. In general, optimal harvesting times are overestimated when size‐homogeneity in the culture is assumed. Our analysis reveals that management predictions are significantly mistaken if the size‐heterogeneity phenomenon is not taken into account.